A064646 -id:A064646 - cp's OEIS frontend

A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

1, 2, 11, 76, 137, 7534, 97943, 1469144, 24975449, 94906706, 9965204131, 229199695012, 5729992375301, 9100576125478, 897316805972131, 563093542209232, 4589775462547450033, 5539384178936577626, 5943759223998947792699, 46361321947191792783052, 9504070999174317520525661

Offset: 1

Ilya Gutkovskiy, May 23 2022

			1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A001147, A053557, A061354, A064646, A103816, A113012, A120265, A143382, A289381, A306858, A354299 (denominators).

S:= 0: R:= NULL:
for n from 1 to 100 do
  S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
  R:= R, numer(S);
od:
R; # Robert Israel, Jan 10 2024

Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Numerator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Numerator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Numerator

Numerators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

A064647 Denominators of partial sums of reciprocals of primorial numbers.

Original entry on oeis.org

2, 3, 10, 105, 770, 15015, 170170, 4849845, 6760390, 3234846615, 200560490130, 72752334655, 4680773285034, 284407855036305, 614889782588491410, 465559406817000639, 44715356980330526490, 19548063559901161830545

Offset: 1

Labos Elemer, Oct 04 2001

			For n = 5, Sum_{j=1..5} 1/A002110(j) = 1/2 + 1/6 + 1/30 + 1/210 + 1/2310 = (1155 + 385 + 77 + 11 + 1)/2310 = 1629/2310 = 543/770, so a(5) = 770.

Cf. A002110, A064646 (numerators).

q[x_] := Apply[Times, Table[Prime[j], {j, 1, x}]]; a[n_] := Denominator[Apply[Plus, Table[1/q[w], {w, 1, n}]]]; Array[a, 18]

list(lim) = {my(s = 0, pr = 1); forprime(p = 1, lim, pr *= p; s += 1/pr; print1(denominator(s), ", "));} \\ Amiram Eldar, Feb 08 2025

A064488 A Beatty sequence: Floor[n*c], where c = A064648 is the sum of the reciprocals of primorials.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 43, 43, 44, 45, 45, 46, 47, 47, 48, 49, 50, 50, 51, 52

Offset: 1

Labos Elemer, Oct 04 2001

nonn

Index entries for sequences related to Beatty sequences

Cf. A002110, A064646-A064648.

With[{nn=80},Floor[Times@@#]&/@Thread[{Range[nn],Accumulate[1/Rest[ FoldList[Times,1,Prime[Range[nn]]]]]}]] (* Harvey P. Dale, Apr 13 2014 *)

cp's OEIS Frontend

A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

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A064647 Denominators of partial sums of reciprocals of primorial numbers.

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A064488 A Beatty sequence: Floor[n*c], where c = A064648 is the sum of the reciprocals of primorials.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica